The Uruguay midfielder moves to Turkey in a deal worth £5.5million, penning a four-year contract with his new club.
Torreira is the seventh player to leave the Emirates this summer.
Torreira wrote on social media following the announcement: "I am very happy and excited to be a new Galatasaray player. I am convinced that this is the beginning of a long and successful history together. Thank you for trusting me and I promise to leave everything for these colours. Let's go Gala!"
Since signing in 2018 for £26m from Sampdoria, Torreira made 89 appearances for the Gunners.